https://file.aiquickdraw.com/mathbot/file/1725559847980bjjgcld0.jpg

The problem in the image involves finding the average cost of production based on the marginal cost function. The marginal cost function provided is:

\[
\frac{dC}{dq} = 0.003q^2 - 0.4q + 40
\]

where $q$ is the number of units produced. We are also given additional information:

- The marginal cost is $27.50 when $q = 50$.
- Fixed costs are $5000.
- We need to find the average cost of producing 100 units.

### Steps to solve:

1. **Integrate the marginal cost function** to find the total cost function $C(q)$.
2. **Use the given condition** $MC(50) = 27.50$ to determine the constant of integration.
3. **Add the fixed cost** to the cost function.
4. **Calculate the total cost** at $q = 100$.
5. **Find the average cost** by dividing the total cost by 100.

Let me go ahead and calculate that.The total cost function, before adding the constant of integration, is:

\[
C(q) = 0.001q^3 - 0.2q^2 + 40q + C_0
\]

where $C_0$ is the constant of integration. Next, we will use the condition that the marginal cost at $q = 50$ is $27.50 to determine $C_0$. Let me calculate that.The given condition that the marginal cost at $q = 50$ is $27.50 holds true, as the marginal cost calculation confirms this value. 

Now, I will determine the constant $C_0$ by considering the fixed costs of $5000 and evaluating the total cost function.The average cost of producing 100 units is $80.00.

Would you like a detailed explanation of any step or further clarification?

Here are five related questions you might find interesting:
1. How do fixed costs influence the total cost function?
2. How can we interpret the marginal cost function in terms of production efficiency?
3. What role does the constant of integration play in calculating total costs?
4. How does average cost differ from marginal cost?
5. What happens to average cost if production continues to increase?

**Tip:** When analyzing cost functions, always differentiate between marginal, total, and average costs, as they provide insights into different aspects of production efficiency.

Costo promedio: Un fabricante ha determinado que la función de costo marginal es dC/dq = 0.003q^2 - 0.4q + 40. Si el costo marginal es de $27.50 cuando q = 50 y los costos fijos son de $5000, ¿cuál es el costo promedio de producir 100 unidades?

Learn how to calculate the average cost of producing 100 units given a marginal cost function. The problem includes integration of the marginal cost function, using fixed costs, and solving for the total cost to find the average cost. Suitable for undergraduate students studying calculus or economics.

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